Selective state space models (SSMs) use a recurrence to mix token information, a process analogous to attention in transformers. By modeling token evolution as an ordinary differential equation and applying input‑to‑state stability, the study proves that SSMs exhibit local exponential stability of consensus equilibria and delineates their domain of attraction for time‑varying weight matrices. Experiments on a pretrained Mamba‑2 model reveal that the output gate controls the degree of consensus, preventing tokens from fully converging.
arXiv:2609.36230v1 Announce Type: new Abstract: We study the dynamical behavior of tokens in transformers from a control-theoretic perspective. Our model includes the feed-forward layer present after...
The paper investigates the training dynamics of attention mechanisms in high-dimensional settings, focusing on attention-indexed models that encompass multi-layer and multi-head architectures. It shows that while the loss landscape can be described by a finite set of trace order parameters, the online stochastic gradient descent dynamics involve an infinite hierarchy of matrix moments that can be accurately approximated by a finite truncated system. The study further reveals that the choice of attention parameterization acts as an implicit bias: untied attention can get trapped in uninformative states, whereas tied attention induces symmetry breaking and enables weak recovery with θ(d² log d) samples, and untied attention exhibits a fast-slow dynamic leading to weak recovery when symmetry is broken.
arXiv:2609.28448v1 Announce Type: cross Abstract: We study the nonequilibrium dynamics of a minimal recurrent transformer with $N$ normalized tokens, $Q=K=I$, and a negative value map $V=-I$. Similar...
arXiv:2602. 19143v2 Announce Type: replace Abstract: This paper studies simple transformers trained on a high-order Markov chain, where the model must incorporate information from multiple past positions, each with different statistical importance.
arXiv:2512. 21113v2 Announce Type: replace Abstract: Transformers are increasingly adopted for modeling and forecasting time-series, yet their internal mechanisms remain poorly understood from a dynamical systems perspective.
arXiv:2606. 24396v1 Announce Type: new Abstract: Large Transformer models function as Dense Associative Memories (DAMs), retrieving knowledge via high-dimensional attractor dynamics driven by the self-attention mechanism \citep{ramsauer2020hopfield, wu2024attention}.
arXiv:2609.39892v1 Announce Type: new Abstract: Looped Transformers repeatedly apply the same set of Transformer layers, giving them a recurrent architecture for latent computation. Their strong perf...
arXiv:2608. 18592v1 Announce Type: new Abstract: Whether distinct neural architectures develop common collective dynamics remains an open question.
arXiv:2608. 08922v1 Announce Type: cross Abstract: Transformer layers generate state-dependent interaction networks: token representations determine the attention matrix, which in turn updates the representations.
arXiv:2609.37921v1 Announce Type: new Abstract: What are the inductive biases of a Transformer architecture? Existing theory on how the forward pass shapes representations either considers whether Tr...
arXiv:2602. 18849v2 Announce Type: replace-cross Abstract: We develop a sensitivity analysis for transformer attention in a geometry aligned with tokenwise computation.
arXiv:2501. 18322v2 Announce Type: replace Abstract: Transformers, which are state-of-the-art in most machine learning tasks, represent the data as sequences of vectors called tokens.